Reduced computation in joint detection

ABSTRACT

A user equipment receives a plurality of transmitted data signals in a communication system. The transmitted data signals are received. A channel response is determined for each received data signal. A system response is determined based on in part the channel signals. The system response is expanded to be piecewise orthogonal. Data from the received data signals is retrieved based on in part the expanded system response.

CROSS REFERENCE TO RELATED APPLICATION

This application is a continuation of U.S. patent application Ser. No.09/662,404 filed Sep. 14, 2000, which in turn claims priority to U.S.Provisional Patent Application No. 60/153,801, filed on Sep. 14, 1999.

BACKGROUND

The invention generally relates to wireless communication systems. Inparticular, the invention relates to joint detection of multiple usersignals in a wireless communication system.

FIG. 1 is an illustration of a wireless communication system 10. Thecommunication system 10 has base stations 12 ₁ to 12 ₅ which communicatewith user equipments (UEs) 14 ₁ to 14 ₃. Each base station 12 ₁ has anassociated operational area where it communicates with UEs 14 ₁ to 14 ₃in its operational area.

In some communication systems, such as code division multiple access(CDMA) and time division duplex using code division multiple access(TDD/CDMA), multiple communications are sent over the same frequencyspectrum. These communications are typically differentiated by theirchip code sequences. To more efficiently use the frequency spectrum,TDD/CDMA communication systems use repeating frames divided into timeslots for communication. A communication sent in such a system will haveone or multiple associated chip codes and time slots assigned to itbased on the communication's bandwidth.

Since multiple communications may be sent in the same frequency spectrumand at the same time, a receiver in such a system must distinguishbetween the multiple communications. One approach to detecting suchsignals is single user detection. In single user detection, a receiverdetects only the communication from a desired transmitter using a codeassociated with the desired transmitter, and treats signals of othertransmitters as interference.

In some situations, it is desirable to be able to detect multiplecommunications simultaneously in order to improve performance. Detectingmultiple communications simultaneously is referred to as jointdetection. Some joint detectors use Cholesky decomposition to perform aminimum mean square error (MMSE) detection and zero-forcing blockequalizers (ZF-BLEs). These detectors have a high complexity requiringextensive receiver resources.

Accordingly, it is desirable to have alternate approaches to jointdetection.

SUMMARY

A user equipment receives a plurality of transmitted data signals in acommunication system. The transmitted data signals are received. Achannel response is determined for each received data signal. A systemresponse is determined based on in part the channel signals. The systemresponse is expanded to be piecewise orthogonal. Data from the receiveddata signals is retrieved based on in part the expanded system response.

BRIEF DESCRIPTION OF THE DRAWING(S)

FIG. 1 is a wireless communication system.

FIG. 2 is a simplified transmitter and a receiver using joint detection.

FIG. 3 is an illustration of a communication burst.

FIG. 4 is an illustration of reduced computation joint detection.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT(S)

FIG. 2 illustrates a simplified transmitter 26 and receiver 28 usingjoint detection in a TDD/CDMA communication system. In a typical system,a transmitter 26 is in each UE 14 ₁ to 14 ₃ and multiple transmittingcircuits 26 sending multiple communications are in each base station 12₁ to 12 ₅. A base station 12 ₁ will typically require at least onetransmitting circuit 26 for each actively communicating UE 14 ₁ to 14 ₃.The joint detection receiver 28 may be at a base station 12 ₁. UEs 14 ₁to 14 ₃ or both. The joint detection receiver 28 receives communicationsfrom multiple transmitters 26 or transmitting circuits 26.

Each transmitter 26 sends data over a wireless communication channel 30.A data generator 32 in the transmitter 26 generates data to becommunicated over a reference channel to a receiver 28. Reference datais assigned to one or multiple codes and/or time slots based on thecommunications bandwidth requirements. A spreading and training sequenceinsertion device 34 spreads the reference channel data and makes thespread reference data time-multiplexed with a training sequence in theappropriate assigned time slots and codes. The resulting sequence isreferred to as a communication burst. The communication burst ismodulated by a modulator 36 to radio frequency. An antenna 38 radiatesthe RF signal through the wireless radio channel 30 to an antenna 40 ofthe receiver 28. The type of modulation used for the transmittedcommunication can be any of those known to those skilled in the art,such as direct phase shift keying (DPSK) or quadrature phase shiftkeying (QPSK).

A typical communication burst 16 has a midamble 20, a guard period 18and two data bursts 22, 24, as shown in FIG. 3. The midamble 20separates the two data bursts 22, 24 and the guard period 18 separatesthe communication bursts to allow for the difference in arrival times ofbursts transmitted from different transmitters. The two data bursts 22,24 contain the communication burst's data and are typically the samesymbol length.

The antenna 40 of the receiver 28 receives various radio frequencysignals. The received signals are demodulated by a demodulator 42 toproduce a baseband signal. The baseband signal is processed, such as bya channel estimation device 44 and a joint detection device 46, in thetime slots and with the appropriate codes assigned to the communicationbursts of the corresponding transmitters 26. The channel estimationdevice 44 uses the training sequence component in the baseband signal toprovide channel information, such as channel impulse responses. Thechannel information is used by the joint detection device 46 to estimatethe transmitted data of the received communication bursts as softsymbols.

The joint detection device 46 uses the channel information provided bythe channel estimation device 44 and the known spreading codes used bythe transmitters 26 to estimate the data of the various receivedcommunication bursts. Although joint detection is described inconjunction with a TDD/CDMA communication system, the same approach isapplicable to other communication systems, such as CDMA.

One approach to joint detection in a particular time slot in a TDD/CDMAcommunication system is illustrated in FIG. 4. A number of communicationbursts are superimposed on each other in the particular time slot, suchas K communication bursts. The K bursts may be from K differenttransmitters. If certain transmitters are using multiple codes in theparticular time slot, the K bursts may be from less than K transmitters.

Each data burst 22, 24 of the communication burst 16 has a predefinednumber of transmitted symbols, such as N_(s). Each symbol is transmittedusing a predetermined number of chips of the spreading code, which isthe spreading factor (SF). In a typical TDD communication system, eachbase station 12 ₁ to 12 ₅ has an associated scrambling code mixed withits communicated data. The scrambling code distinguishes the basestations from one another. Typically, the scrambling code does notaffect the spreading factor. Although the terms spreading code andfactor are used hereafter, for systems using scrambling codes, thespreading code for the following is the combined scrambling andspreading codes. Each data burst 22, 24 has N_(s)×SF chips.

The joint detection device 46 estimates the value that each data burstsymbol was originally transmitted. Equation 1 is used to determine theunknown transmitted symbols.r=Ad+n  Equation 1

In Equation 1, the known received combined chips, r, is a product of thesystem response, A, and the unknown transmitted symbols, d. The term, n,represents the noise in the wireless radio channel.

For K data bursts, the number of data burst symbols to be recovered isNs×K. For analysis purposes, the unknown data burst symbols are arrangedinto a column matrix, d. The d matrix has column blocks, d₁ to d_(Ns),of unknown data symbols. Each data symbol block, d_(i), has the i^(th)unknown transmitted data symbol in each of the K data bursts. As aresult, each column block, d_(i), has K unknown transmitted symbolsstacked on top of each other. The blocks are also stacked in a column ontop of each other, such that d₁ is on top of d₂ and so on.

The joint detection device 46 receives a value for each chip asreceived. Each received chip is a composite of all K communicationbursts. For analysis purposes, the composite chips are arranged into acolumn matrix, r. The matrix r has a value of each composite chip,totaling Ns*SF chips.

A is the system response matrix. The system response matrix, A, isformed by convolving the impulse responses with each communication burstchip code. The convolved result is rearranged to form the systemresponse matrix, A (step 48).

The joint detection device 46 receives the channel impulse response,h_(i), for each i^(th) one of the K communication bursts from thechannel estimation device 44. Each h_(i) has a chip length of W. Thejoint detection device convolves the channel impulse responses with theknown spreading codes of the K communication bursts to determine thesymbol responses, s₁to s_(K), of the K communication bursts. A commonsupport sub-block, S, which is common to all of the symbol responses isof length K×(SF+W−1).

The A matrix is arranged to have Ns blocks, B₁ to B_(Ns). Each block hasall of the symbol responses, s₁ to s_(K), arranged to be multiplied withthe corresponding unknown data block in the d matrix, d₁ to d_(Ns). Forexample, d₁ is multiplied with B₁. The symbol responses, s₁ to s_(K),form a column in each block matrix, B_(i), with the rest of the blockbeing padded with zeros. In the first block, B₁, the symbol response rowstarts at the first row. In the second block, the symbol response row isSF rows lower in the block and so on. As a result, each block has awidth of K and a height of Ns×SF. Equation 2 illustrates an A blockmatrix showing the block partitions. $\begin{matrix}\begin{matrix}{A = \begin{bmatrix}{\underset{\_}{s}}_{1} & {\underset{\_}{s}}_{2} & \cdots & {\underset{\_}{s}}_{K} & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & \cdots \\0 & 0 & 0 & 0 & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \quad \\\quad & \quad & \quad & \quad & 0 & 0 & 0 & 0 & \quad & \quad & \quad & \quad & \quad \\\quad & \quad & \quad & \quad & {\underset{\_}{s}}_{1} & {\underset{\_}{s}}_{2} & \cdots & {\underset{\_}{s}}_{K} & \quad & \quad & \quad & \quad & \quad \\\quad & \quad & \quad & \quad & 0 & 0 & 0 & 0 & \quad & \quad & \quad & \quad & \quad \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & 0 & 0 & 0 & 0 & {\cdots\quad} \\\quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & {\underset{\_}{s}}_{1} & {\underset{\_}{s}}_{2} & \cdots & {\underset{\_}{s}}_{K} & \quad \\\quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & 0 & 0 & 0 & 0 & \quad \\\quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \vdots & \vdots & \vdots & \vdots & \quad \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots\end{bmatrix}} \\{= \lbrack \begin{matrix}B_{1} & B_{2} & \cdots &  B_{N_{s}} \rbrack\end{matrix} }\end{matrix} & {{Equation}\quad 2}\end{matrix}$

The n matrix has a noise value corresponding to each received combinedchip, totaling Ns×SF chips. For analysis purposes, the n matrix isimplicit in the received combined chip matrix, r.

Using the block notation, Equation 1 can be rewritten as Equation 3.$\begin{matrix}{\underset{\_}{r} = {{\begin{bmatrix}B_{1} & B_{2} & B_{3} & \cdots & { B_{N_{s}} \rbrack \times \lbrack \begin{matrix}\underset{\_}{d_{1}} \\\underset{\_}{d_{2}} \\\underset{\_}{d_{3}} \\\vdots \\\underset{\_}{d_{N_{s}}}\end{matrix} }\end{bmatrix} + \underset{\_}{n}}\quad = {{\sum\limits_{i = 1}^{N_{s}}\quad{B_{i}\underset{\_}{d_{i}}}} + \underset{\_}{n}}}} & {{Equation}\quad 3}\end{matrix}$

Using a noisy version of the r matrix, the value for each unknown symbolcan be determined by solving the equation. However, a brute forceapproach to solving Equation 1 requires extensive processing.

To reduce the processing, the system response matrix, A, isrepartitioned. Each block, B_(i), is divided into Ns blocks having awidth of K and a height of SF. These new blocks are referred to as A₁ toA_(L) and 0. L is the length of the common support S, as divided by theheight of the new blocks, A₁ to A_(L), per Equation 4. $\begin{matrix}{L = \lceil \frac{{SF} + W - 1}{SF} \rceil} & {{Equation}\quad 4}\end{matrix}$

Blocks A₁ to A_(L) are determined by the supports, s₁ to s_(K), and thecommon support, S. A 0 block is a block having all zeros. Arepartitioned matrix for a system having a W of 57, SF of 16 and an L of5 is shown in Equation 5. $\begin{matrix}{A = {\quad\begin{bmatrix}A_{1} & 0 & 0 & 0 & 0 & 0 & 0 & \ldots & 0 & 0 & 0 & 0 \\A_{2} & A_{1} & 0 & 0 & 0 & 0 & 0 & \ldots & 0 & 0 & 0 & 0 \\A_{3} & A_{2} & A_{1} & 0 & 0 & 0 & 0 & \ldots & 0 & 0 & 0 & 0 \\A_{4} & A_{3} & A_{2} & A_{1} & 0 & 0 & 0 & \ldots & 0 & 0 & 0 & 0 \\A_{5} & A_{4} & A_{3} & A_{2} & A_{1} & 0 & 0 & \ldots & 0 & 0 & 0 & 0 \\0 & A_{5} & A_{4} & A_{3} & A_{2} & A_{1} & 0 & \ldots & 0 & 0 & 0 & 0 \\0 & 0 & A_{5} & A_{4} & A_{3} & A_{2} & A_{1} & ⋰ & 0 & 0 & 0 & 0 \\0 & 0 & 0 & A_{5} & A_{4} & A_{3} & A_{2} & ⋰ & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & A_{5} & A_{4} & A_{3} & ⋰ & A_{1} & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & A_{5} & A_{4} & ⋰ & A_{2} & A_{1} & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & A_{5} & ⋰ & A_{3} & A_{2} & A_{1} & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋰ & A_{4} & A_{3} & A_{2} & A_{1}\end{bmatrix}}} & {{Equation}\quad 5}\end{matrix}$

To reduce the complexity of the matrix, a piecewise orthogonalizationapproach is used. Any of the blocks B_(i) for i being L or greater isnon-orthogonal to any of the preceding L blocks and orthogonal to anyblocks preceding by more than L. Each 0 in the repartitioned A matrix isan all zero block. As a result to use a piecewise orthogonalization, theA matrix is expanded (step 50).

The A matrix is expanded by padding L−1 zero blocks to the right of eachblock of the A matrix and shifting each row in the A matrix by its rownumber less one. To illustrate for the A1 block in row 2 of FIG. 2, four(L-1) zeros are inserted between A2 and A1 in row 2. Additionally, blockA1 (as well as A2) is shifted to the right by one column (row 2−1). As aresult, Equation 5 after expansion would become Equation 6.$\begin{matrix}{A_{\exp} = {\quad\begin{bmatrix}A_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \quad \\0 & A_{2} & 0 & 0 & 0 & A_{1} & 0 & 0 & 0 & 0 & 0 & \quad \\0 & 0 & A_{3} & 0 & 0 & 0 & A_{2} & 0 & 0 & 0 & A_{1} & \quad \\0 & 0 & 0 & A_{4} & 0 & 0 & 0 & A_{3} & 0 & 0 & 0 & \quad \\0 & 0 & 0 & 0 & A_{5} & 0 & 0 & 0 & A_{4} & 0 & 0 & \quad \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & A_{5} & 0 & \cdots \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \quad \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \quad \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \quad \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \quad \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \quad \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \quad\end{bmatrix}}} & {{Equation}\quad 6}\end{matrix}$

To accommodate the expanded A matrix, the d matrix must also beexpanded, d_(exp). Each block, d₁ to d_(Ns), is expanded to a new block,d_(exp1) to d_(expNs). Each expanded block, d_(exp1) to d_(expNs), isformed by repeating the original block L times. For example ford_(exp1), a first block row would be created having L versions of d1,stacked one below the other.

As a result, Equation 1 can be rewritten as Equation 7. $\begin{matrix}\begin{matrix}{\underset{\_}{r} = {{A_{e\quad{xp}} \cdot \underset{\_}{d_{e\quad{xp}}}} + \underset{\_}{n}}} \\{= {{\begin{matrix}\lbrack B_{exp1}  & B_{exp2} & B_{exp3} & \cdots & { B_{{expN}_{s}} \rbrack \times}\end{matrix}\begin{bmatrix}\underset{\_}{d_{exp1}} \\\underset{\_}{d_{exp2}} \\\underset{\_}{d_{exp3}} \\\vdots \\\underset{\_}{d_{{expN}_{s}}}\end{bmatrix}} +}} \\{{\underset{\_}{n} = {{\sum\limits_{i = 1}^{N_{s}}\quad{B_{expi}\underset{\_}{d_{expi}}}} + \underset{\_}{n}}},}\end{matrix} & {{Equation}\quad 7}\end{matrix}$Equation 7 can be rewritten to partition each B_(expi) orthogonally in Lpartitions, U_(j) ^((I)), j=1 to L, as in Equation 8. $\begin{matrix}\begin{matrix}{\underset{\_}{r} = {{A_{e\quad{xp}} \cdot \underset{\_}{d_{e\quad{xp}}}} + \underset{\_}{n}}} \\{= {{\sum\limits_{i = 1}^{N_{s}}\quad{\begin{matrix}\lbrack U_{1}^{(i)}  & U_{1}^{(i)} & \cdots & { U_{L}^{(i)} \rbrack \times}\end{matrix}\begin{bmatrix}\underset{\_}{d_{i}} \\\underset{\_}{d_{i}} \\\underset{\_}{d_{i}} \\\vdots \\\underset{\_}{d_{i}}\end{bmatrix}}} +}} \\{\underset{\_}{n} = {{\sum\limits_{i = 1}^{N_{s}}\quad{\sum\limits_{j = 1}^{L}{U_{j}^{(i)}\underset{\_}{d_{i}}}}} = {{\sum\limits_{i = 1}^{N_{s}}{B_{i}\underset{\_}{d_{i}}}} + \underset{\_}{n}}}}\end{matrix} & {{Equation}\quad 8}\end{matrix}$

To reduce computational complexity, a QR decomposition of the A_(exp)matrix is performed (step 52). Equation 9 illustrates the QRdecomposition of A_(exp).A _(exp) =Q _(exp) R _(exp)  Equation 9Due to the orthogonal partitioning of A_(exp), the QR decomposition ofA_(exp) is less complex. The resulting Q_(exp) and R_(exp) matrices areperiodic with an initial transient extending over L blocks. Accordingly,Q_(exp) and R_(exp) can be determined by calculating the initialtransient and one period of the periodic portion. Furthermore, theperiodic portion of the matrices is effectively determined byorthogonalizing A₁ to A_(L). One approach to QR decomposition is aGramm-Schmidt orthogonalization.

To orthogonalize A_(exp) as in Equation 6, B_(exp1) is othogonalized byindependently orthogonalizing each of its orthogonal partitions, {U_(j)^((i))}, j=1 . . . L. Each {A_(j)}, j=1 . . . L is independentlyorthogonalized, and the set is zero-padded appropriately. {Q_(j)} arethe orthonormal sets obtained by orthogonalizing {U_(j) ^((i))}. Todetermine B_(exp2), its U₁ ⁽²⁾ needs to be orthogonalized with respectto only Q₂ of B_(exp1) formed previously. U₂ ⁽²⁾, U₃ ⁽²⁾ and U₄ ⁽²⁾ onlyneed to be orthogonalized with respect to only Q₃, Q₄ and Q₅,respectively. U₅ ⁽²⁾ needs to be ortogonalized to all previous Qs andits orthogonalized result is simply a shifted version of Q₅ obtainedfrom orthogonalizing B_(exp1).

As the orthogonalizing continues, beyond the initial transient, thereemerges a periodicity which can be summarized as follows. The result oforthogonalizing B_(exp1), i≧6 can be obtained simply by a periodicextension of the result of orthogonalizing B_(exp5).

The orthogonalization of B_(exp5), is accomplished as follows. Its Q₅ isobtained by orthogonalizing A₅, and then zero padding. Its Q₄ isobtained by orthogonalizing the support of Q₅ and A₄, [sup(Q₅) A₄], andthen zero padding. Since sup(Q₅) is already an orthogonal set, only A₄needs to be othogonalized with respect to sup(Q₅) and itself. Its Q₃ isobtained by orthogonalizing [sup(Q₅) sup(Q₄) A₃] and then zero padding.Its Q₂ is obtained by orthogonalizing [sup(Q₅) sup(Q₄) sup(Q₃) A₂] andthen zero padding. Its Q₁ is obtained by orthogonalizing [sup(Q₅)sup(Q₄) sup(Q₃) sup(Q₂) A₁] and then zero padding. Apart from theinitial transient, the entire A_(exp) can be efficiently orthogonalized,by just orthogonalizing A_(p) per Equation 10.A _(p) =[A ₅ A ₄ A ₃ A ₂ A ₁]  Equation 10By effectively orthogonalizing the periodic portion of A_(exp) by usingonly A_(p), computational efficiency is achieved. Using a more compactnotation, Q_(i) ^(s), for sup (Q_(i)), this orthogonalization of A_(p)results in the orthonormal matrix, Q_(p), of Equation 11.Q _(p) =[Q ₅ ^(s) Q ₄ ^(s) Q ₃ ^(s) Q ₂ ^(s) Q ₁ ^(s)]  Equation 11The periodic part of Q_(exp) is per Equation 12. $\begin{matrix}{{PeriodicPartofQ}_{e\quad{xp}} = {\quad\begin{bmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\quad\cdots} \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \quad \\Q_{1}^{s} & 0 & 0 & 0 & 0 & \quad & 0 & 0 & 0 & 0 & \quad \\0 & Q_{2}^{s} & 0 & 0 & 0 & Q_{1}^{s} & 0 & 0 & 0 & 0 & \quad \\0 & 0 & Q_{3}^{s} & 0 & 0 & 0 & Q_{2}^{s} & 0 & 0 & 0 & \quad \\0 & 0 & 0 & Q_{4}^{s} & 0 & 0 & 0 & Q_{3}^{s} & 0 & 0 & \quad \\0 & 0 & 0 & 0 & Q_{5}^{s} & 0 & 0 & 0 & Q_{4}^{s} & 0 & {\quad\cdots} \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & Q_{5}^{s} & \quad \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \quad \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & 0 & \quad \\\quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\cdots\quad}\end{bmatrix}}} & {{Equation}\quad 12}\end{matrix}$

To constructing the upper triangular matrix R_(exp),<A_(i)>_(j) is ablock of size K×K representing the projections of each column of A_(i)onto all the columns of Q_(j) ^(s). For example, the first column of<A₄>₅ represents the projections of the first column of A₄ on each ofthe K columns of Q₅ ^(s). Similarly, <A₄>₄ represents the projections ofthe first column of A₄ on each of the K columns of Q₄ ^(s). However,this block will be upper triangular, because the k^(th) column of A₄belongs to the space spanned by the orthonormal vectors of Q₅ ^(s) andthe first k vectors of Q₄ ^(s). This block is also orthogonal tosubsequent vectors in Q₄ ^(s), leading to an upper triangular <A₄>₄. Any<A_(i)>_(j) with i=j will be upper triangular. To orthogonalize otherblocks, the following results.

The first block of B_(exp5), viz., U₁ ⁽⁵⁾ is formed by a linearcombination of {Q_(j) ^(s)}, j=1 . . . 5, with coefficients given by<A₁>_(j), j=1 . . . 5. The second block, U₂ ⁽⁵⁾, is formed by a linearcombination of {Q_(j) ^(s)}, j=2 . . . 5, with coefficients given by<A₂>_(j), j=2 . . . 5. The third block, U₃ ⁽⁵⁾, is formed by a linearcombination of {Q_(j) ^(s)}, j=3 . . . 5, with coefficients given by<A₂>_(j), j=3 . . . 5. The fourth block, U₄ ⁽⁵⁾, is formed by a linearcombination of {Q_(j) ^(s)}, j=4,5, with coefficients given by <A₂>_(j),j=4,5. The fifth block, U₅ ⁽⁵⁾, is formed by Q₅ ^(s)×<A₅>₅.

Accordingly, the coefficients in the expansion of subsequent B_(expi),i≧6 are simply periodic extensions of the above. Since the R_(exp)entries are computed during the orthogonalization of A_(exp), noadditional computations are needed to construct R_(exp). Disregardingthe initial transient, the remainder of R_(exp) is periodic, and twoperiods of it are shown in Equation 13. $\begin{matrix}{R_{e\quad{xp}} = \begin{bmatrix}\quad & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots \\\quad & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \quad & \quad \\\quad & 0 & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad \\\quad & \langle A_{1} \rangle_{5} & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad \\\quad & 0 & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad \\\quad & 0 & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad \\\quad & 0 & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad & \quad \\\quad & \langle A_{1} \rangle_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \quad & \quad \\\quad & 0 & \langle A_{2} \rangle_{5} & 0 & 0 & 0 & \langle A_{1} \rangle_{5} & 0 & 0 & 0 & \quad & \quad \\\quad & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \quad & \quad \\\quad & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \quad & \quad \\\quad & \langle A_{1} \rangle_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \quad & \quad \\\quad & 0 & \langle A_{2} \rangle_{4} & 0 & 0 & 0 & \langle A_{1} \rangle_{4} & 0 & 0 & 0 & \quad & \quad \\\quad & 0 & 0 & \langle A_{3} \rangle_{5} & 0 & 0 & 0 & \langle A_{2} \rangle_{5} & 0 & 0 & \quad & \quad \\\quad & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \quad & \quad \\\quad & \langle A_{1} \rangle_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \quad & \quad \\0 & 0 & \langle A_{2} \rangle_{3} & 0 & 0 & 0 & \langle A_{1} \rangle_{3} & 0 & 0 & 0 & \quad & \quad \\\quad & 0 & 0 & \langle A_{3} \rangle_{4} & 0 & 0 & 0 & \langle A_{2} \rangle_{4} & 0 & 0 & \quad & \quad \\\quad & 0 & 0 & 0 & \langle A_{4} \rangle_{5} & 0 & 0 & 0 & \langle A_{3} \rangle_{5} & 0 & \quad & {\cdots\quad} \\\quad & \langle A_{1} \rangle_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \quad & \quad \\\quad & 0 & \langle A_{2} \rangle_{2} & 0 & 0 & 0 & \langle A_{1} \rangle_{2} & 0 & 0 & 0 & \quad & \quad \\\quad & 0 & 0 & \langle A_{3} \rangle_{3} & 0 & 0 & 0 & \langle A_{2} \rangle_{3} & 0 & 0 & \quad & \quad \\\quad & 0 & 0 & 0 & \langle A_{4} \rangle_{4} & 0 & 0 & 0 & \langle A_{3} \rangle_{4} & 0 & \quad & \quad \\\quad & 0 & 0 & 0 & 0 & \langle A_{5} \rangle_{5} & 0 & 0 & 0 & \langle A_{4} \rangle_{5} & \quad & \quad \\\quad & 0 & 0 & 0 & 0 & 0 & \langle A_{1} \rangle_{1} & 0 & 0 & 0 & \quad & \quad \\\quad & 0 & 0 & 0 & 0 & 0 & 0 & \langle A_{2} \rangle_{2} & 0 & 0 & \quad & \quad \\\quad & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \langle A_{3} \rangle_{3} & 0 & \quad & \quad \\\quad & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \langle A_{4} \rangle_{4} & \quad & \quad \\\quad & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \langle A_{5} \rangle_{5} & \quad \\\quad & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \langle A_{1} \rangle_{1}\end{bmatrix}} & {{Equation}\quad 16}\end{matrix}$The least squares approach to solving Q_(exp) and R_(exp) is shown inEquation 14.Q _(exp) ·R _(exp) ·d _(exp) =r  Equation 14By pre-multiplying both sides of Equation 14 by the transpose ofQ_(exp), Q_(exp) ^(T), and using Q_(exp) ^(T)·Q_(exp)=I_(LKN) _(s) ,Equation 14 becomes Equation 15.R _(exp) ·d _(exp) =Q _(exp) ^(T) r  Equation 15Equation 15 represents a triangular system whose solution also solvesthe LS problem of Equation 14.

Due to the expansion, the number of unknowns is increased by a factor ofL. Since the unknowns are repeated by a factor of L, to reduce thecomplexity, the repeated unknowns can be collected to collapse thesystem. R_(exp) is collapsed using L coefficient blocks, CF₁ to CF_(L),each having a width and a height of K. For a system having an L of 5,CF₁ to CF₅ can be determined as in Equation 16.CF ₁ =<A ₁>₁ +<A ₂>₂ +<A ₃>₃ +<A ₄>₄ +<A ₅>₅CF ₂ =<A ₁>₂ +<A ₂>₃ +<A ₃>₄ +<A ₄>₅CF ₃ =<A ₁>₃ +<A ₂>₄ +<A ₃>₅CF ₄ =<A ₁>₄ +<A ₂>₅CF ₅ =<A ₁>₅  Equation 16Collapsing R_(exp) using the coefficient blocks produces a Cholesky likefactor, Ĝ (step 54). By performing analogous operations on the righthand side of Equation 15 results in a banded upper triangular system ofheight and width of K×Ns as in Equation 17. $\quad\begin{matrix}{{\begin{bmatrix}{Tr}_{1} & {Tr}_{2} & {Tr}_{3} & {Tr}_{4} & {CF}_{5} & 0 & 0 & 0 & 0 & 0 & \cdots \\0 & {Tr}_{1} & {Tr}_{2} & {CF}_{3} & {CF}_{4} & {CF}_{5} & 0 & 0 & 0 & 0 & \cdots \\0 & 0 & {CF}_{1} & {CF}_{2} & {CF}_{3} & {CF}_{4} & {CF}_{5} & 0 & 0 & 0 & \cdots \\0 & 0 & 0 & {CF}_{1} & {CF}_{2} & {CF}_{3} & {CF}_{4} & {CF}_{5} & 0 & 0 & \cdots \\0 & 0 & 0 & 0 & {CF}_{1} & {CF}_{2} & {CF}_{3} & {CF}_{4} & {CF}_{5} & 0 & \cdots \\\vdots & \vdots & \vdots & \vdots & 0 & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & 0\end{bmatrix} \times \begin{bmatrix}\underset{\_}{d_{1}} \\\underset{\_}{d_{2}} \\\underset{\_}{d_{3}} \\\vdots \\\underset{\_}{d_{N_{s}}}\end{bmatrix}} = \underset{\_}{\hat{r}}} & {{{Equation}\quad 17}\quad}\end{matrix}$Tr₁ to Tr₄ are the transient terms and {circumflex over (r)}. By solvingthe upper triangle via back substitution, Equation 17 can be solved todetermine d (step 56). As a result, the transmitted data symbols of theK data bursts is determined.

Using the piecewise orthogonalization and QR decomposition, thecomplexity of solving the least square problem when compared with abanded Cholesky decomposition is reduced by a factor of 6.5.

1. A user equipment for receiving a plurality of transmitted datasignals in a communication system, the user equipment comprising: anantenna for receiving the transmitted data signals; a channel estimationdevice for determining a channel response for each received data signal;and a joint detection device having an input configured to receive thechannel responses and the received data signals for determining a systemresponse based on in part the channel signals, expanding the systemresponse to be piecewise orthogonal, and retrieving data from thereceived data signals based on in part the expanded system response. 2.The user equipment of claim 1 for use in a time division duplex usingcode division multiple access communication system.
 3. The userequipment of claim 1 wherein each of the transmitted data signals has anassociate code and is transmitted in a shared frequency spectrum and thesystem response is determined by convolving the associated chip codeswith the channel response.
 4. A user equipment for receiving a pluralityof transmitted data signals in a communication system, the userequipment comprising: means for receiving the transmitted data signals;means for determining a channel response for each received data signal;and means for determining a system response based on in part the channelsignals; means for expanding the system response to be piecewiseorthogonal, and means for retrieving data from the received data signalsbased on in part the expanded system response.
 5. The user equipment ofclaim 4 for use in a time division duplex using code division multipleaccess communication system.
 6. The user equipment of claim 4 whereineach of the transmitted data signals has an associate code and istransmitted in a shared frequency spectrum and the system response isdetermined by convolving the associated chip codes with the channelresponse.